Find a solution to the equation $x^2-2x-2007y^2=0$ in positive integers.

Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.

The number $10^{2007}$ is written on the blackboard. Anne and Berit play a two player game in which the player in turn performs one of the following operations: 1) replace a number $x$ on the blackboard with two integers $a,b>1$ such that $ab=x$. 2) strike off one or both of two equal numbers on the blackboard. The person who cannot perform any operation loses. Who has the winning strategy if Anne starts?

A line through $A$ intersects a circle at points $B,C$ with $B$ between $A,C$. The two tangents from $A$ intersect the circle at $S,T$. $ST$ and $AC$ intersect at $P$. Show that $\frac{AP}{PC}=2\frac{AB}{BC}$.